The roughness height and drag law over the water surface based on the hypothesis of local equilibrium

Abstract

A series of measurements of winds and wind-waves were carried out in wind-wave flumes. A data analysis based on the hypothesis of local equilibrium yielded a new empirical formula on the controversial quantity of roughness heightz 0 over the water surface: $$\tilde z_0 = 1.02 x 10^{ - 2} \tilde u$$ , where the nondimensional roughness height $$\tilde z_0 $$ is defined bygz 0/u * 2 and the wave-wind parameterũ byω p u */g, g being the gravitational acceleration,u * the friction velocity of air,ω p the peak frequency of wind-wave spectra. The obtained formula is compared with Charnock's (1955) and Toba's (1979) proposals; $$\tilde z_0 $$ is constant in the former and inversely proportional toũ in the latter. As in Toba's, this formula immediately leads to a practically important conclusion that the drag coefficientC d depends not merely on the usual variableU 10 (wind velocity at 10m height over the water surface), but also on the surface state represented by wind-waves. An explicit expression is provided for the drag coefficient incorporating the wave-wind parameter; it covers the range ofC d calculated from most of the previous drag formulas, by varying the wave-wind parameter.